Assignment 6 Solution

Part 1: Detecting Image Contrast Enhancement




In many scenarios, it is important for a forensic investigator to understand how a digital image has been pro-cessed. One family of image processing operations which we studied earlier in this course is contrast enhancement operations. These operations apply a nonlinear mapping to the pixel values of a digital image to e ectively increase they dynamic range of pixel values. This has the e ect of increasing the visual contrast of a digital image.




In class, we examined how contrast enhancement operations introduce ngerprints into a digital image’s pixel value histogram. For a number of reasons, we can model the pixel value histogram of an unaltered digital image as some \smooth" function. Contrast enhancement ngerprints take the form of impulsive peaks and gaps into an image’s pixel value histogram.




Visually examine the pixel value histograms of the images imageCE1.tif, imageCE2.tif, imageCE3.tif, and imageCE4.tif. Based o of your knowledge of contrast enhancement ngerprints, which of these images are likely to have been contrast enhanced?




Note: It may be useful to use the Matlab command imhist to calculate the image’s pixel value histogram. When plotting the histogram, however, it is often more useful to use the bar functin to display the histogram rather than imhist which displays the histogram as a stem plot.




Examine the pixel value histograms of the unaltered images unaltIm1.tif, unaltIm2.tif, and unaltIm3.tif. Next, apply gamma correction with = 0:7 to each of these images and examine the pixel value histograms of the gamma corrected images. Repeat this procedure again, but this time modify each of the unaltered images using with = 1:3.




What in uence does the unaltered image’s pixel value histogram have on the location and amplitude of the contrast enhancement ngerprints in each of the gamma corrected images. What in uence does the contrast enhancement mapping (in this case speci ed by the value of ) have on the location and amplitude of the contrast enhancement ngerprints in each of the gamma corrected images. Include the pixel value histograms of each of the unaltered and contrast enhanced images in your report.




In class, we discussed the unique artifacts introduced into a \smooth" pixel value histogram by both con-tractive and expansive mappings. The image imageCE5.tif has been contrast enhanced using gamma correction. Examine its pixel value histogram and identify which regions of the contrast enhancement map-ping are locally expansive and which are locally contractive. Based o of this information, is greater than 1 or less than 1?




Part 2: Detecting Image Resampling and Resizing




Image resampling operations are another family of image processing operations which a forensic investigator may wish to detect. Most commonly, image resampling occurs when a digital image is resized. In class, we discussed how image resampling ngerprints can be detected using the method proposed by Popescu and Farid which makes use of the Expectation-Maximization (EM) algorithm, as well as the more computationally e cient algorithm proposed by Kirchner.




Rather than making use of the computationally expensive EM algorithm to estimate the relationship between pixels, Kirchner’s resampling detection algorithm uses a xed linear prediction lter to approximate this rela-tionship. Kirchner was able to demonstrate that the variance of the resulting prediction error will be periodic,




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ECES 435 - Drexel University




Instructor: Matthew Stamm







regardless of the linear prediction lter used. As a result, an approximation of an image’s p-map can be computed without needing to use the EM algorithm to calculate the optimal prediction lter.




Kirchner’s algorithm for e ciently approximating an image’s p-map can be summarized as follows:

^

1. Obtain a prediction I(x; y) of the value of each pixel on the basis of its neighboring pixel values by ltering the image I(x; y) using the linear prediction lter



= 2
0:50


0


00:50:


3
(1)


4
0:25
0:50
25




0:25
0:50
0:25 5


2.
Calculate the resulting prediction error e(x; y) according to the formula e(x; y) = I(x; y)
^
I(x; y).
3.
Calculate the approximate p-map p using the equation














p(x; y) = exp


e(
x; y)
:
(2)













Write a Matlab function that uses Kirchner’s algorithm to calculate an image’s p-map. This function should accept the image to be examined as an input and a matrix corresponding to the image’s p-map as the output. When calculating the image’s p-map, set = 1, = 2, and = 1. Please fully comment your code and append it to your report.




Use your Matlab function to calculate the p-maps of the images resampIm1.tif, resampIm2.tif, re-sampIm3.tif, and resampIm4.tif. Visually examine these p-maps and include them in your report. Which of these p-maps exhibit periodic properties that are indicative of image resampling?




Note: It may be useful to use the function imagesc to display each image’s p-map.




Use the Matlab function showFreqPmap provided on the course’s Blackboard page to plot the magnitude of the Fourier transform of the image’s p-map. Which of these plots contain resampling ngerprints? Include the plot of the magnitude of the Fourier transform of each image’s p-map in your report. If contrast enhancement ngerprints are present in an image, clearly label these ngerprints.












































































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